Does gravity turn time into space?

I am preparing a simple demonstration of gravity for a lay audience, inspired by this applet and the well-known book Relativity Visualized. These resources suggest that for a free-falling object, the curvature of spacetime "converts" some of the object's travel through time into travel through space. That seems to be the case: As the object accelerates due to gravity, we fixed observers see its clock slow down (neglecting GR time dilation caused by the object approaching the gravitational mass).

In my presentation, to explain why a released object goes from "stationary" to falling, I'd like to say something to that effect (with graphics similar to the above applet): that when spacetime is curved, a free-falling object's ordinary travel through time spontaneously begins to transform into travel through space as well.

Is the statement reasonably accurate? Mind you, I am merely trying to create a qualitative "aha!" moment, not a rigorous explanation. Thank you.